Reports of the Academy of Sciences of the USSR

1965, Volume 160, No. 5

MATHEMATICS

A. Guichardet (France)

ON TENSOR PRODUCTS OF \(C^*\)-ALGEBRAS

(Presented by Academician P. S. Novikov on 21 IX 1964)

1. If \(A_1\) and \(A_2\) are \(C^*\)-algebras, then their algebraic tensor product is denoted by \(A_1 \otimes A_2\); on \(A_1 \otimes A_2\) one can define the norm

\[ \|x\|_*=\sup_{\pi_1,\pi_2}\|(\pi_1\otimes \pi_2)(x)\|, \]

where \(\pi_i\) is an arbitrary representation of \(A_i\). This norm has been studied in \((^{1-4})\); it has the following properties

\[ \begin{gathered} \|xy\|_* \leq \|x\|_* \|y\|_*,\\ \|xx^*\|_*=\|x\|_*^2,\\ \|x_1\otimes x_2\|_*=\|x_1\|\,\|x_2\|_*,\\ \|x\|_*=\|(\rho_1\otimes \rho_2)(x)\|, \end{gathered} \tag{1} \]

where \(\rho_i\) is a faithful representation of \(A_i\); the completion of \(A_1\otimes A_2\) in this norm is a \(C^*\)-algebra, which we denote by \(A_1\otimes A_2\); it is known \((^2)\) that \(\|\ \|_*\) is the least norm satisfying (1). Here another norm on \(A_1\otimes A_2\) will be defined and a new completion studied.

2. To every pair \((\pi_1,\pi_2)\) of representations of \(A_1\) and \(A_2\) in one Hilbert space satisfying

\[ \pi_1(x_1)\cdot \pi_2(x_2)=\pi_2(x_2)\cdot \pi_1(x_1),\qquad x_i\in A_i, \]

one can associate a representation \(\pi\) of the algebra \(A_1\otimes A_2\) such that

\[ \pi(x_1\otimes x_2)=\pi_1(x_1)\cdot \pi_2(x_2), \tag{2} \]

\[ \|\pi(x_1\otimes x_2)\|\leq \|x_1\|\cdot\|x_2\|; \tag{3} \]

conversely, every representation of \(A_1\otimes A_2\) satisfying (3) is obtained in this way (this is proved as Proposition 1 in \((^1)\)).

Definition. For every \(x\in A_1\otimes A_2\), put

\[ \|x\|_\nu=\sup_{\pi}\|\pi(x)\|, \]

where \(\pi\) is an arbitrary representation of \(A_1\otimes A_2\) satisfying (3).

Theorem 1.
1) The function \(\|\ \|_\nu\) is the greatest norm on \(A_1\otimes A_2\) having properties analogous to (1).
2) If \(A_1\) or \(A_2\) is of type I, then \(\|\ \|_\nu=\|\ \|_*\).

It is not hard to see that if

\[ x=\sum_{i=1}^{n} x_{1,i}\otimes x_{2,i}\in A_1\otimes A_2, \]

then

\[ \|x\|_*\leq \|x\|_\nu\leq \sum_{i=1}^{n}\|x_{1,i}\|\,\|x_{2,i}\| \]

and that \(\|\ \|_\nu\) is a norm with properties (1).

Let now \(p\) be a norm with properties (1); realize the completion \(B\) with respect to the norm \(p\) in a Hilbert space; the inequality \(p(x)\leq \|x\|_{\nu}\) follows from the fact that the natural mapping \(A_1\otimes A_2\to B\) is a representation with properties (3). Assertion 2) follows from Theorem 3 \((^2)\). We shall denote by \(A_1\overset{\vee}{\otimes}A_2\) the \(C^*\)-algebra which is the completion of \(A_1\otimes A_2\) with respect to the norm \(\|\ \|_{\nu}\); the identity mapping extends to a homomorphism \(A_1\overset{\vee}{\otimes}A_2\) onto \(A_1\overset{*}{\otimes}A_2\), which we shall call canonical.

Remark 1. It is not known whether the norms \(\|\ \|_*\) and \(\|\ \|_{\nu}\) always coincide; proving this would be equivalent to proving the following assertion: if \(T_1,\ldots,T_n,S_1,\ldots,S_n\) are operators in a Hilbert space \(H\) and \(S_iT_j=T_jS_i\), then

\[ \left\|\sum_{i=1}^{n} S_iT_i\right\|\leq \left\|\sum_{i=1}^{n} S_i\otimes T_i\right\|, \]

where \(S_i\otimes T_i\) denotes an operator in \(H\otimes H\). It is known only that this assertion is always valid when \(T_i\) belongs to some \(C^*\)-algebra \(A\) of type I and \(S_i\in A'\).

3. Tensor products of homomorphisms. Let \(u_i\), \(i=1,2\), be a homomorphism of the algebra \(A_i\) into the \(C^*\)-algebra \(B_i\); the homomorphism \(u_1\otimes u_2\) of \(A_1\otimes A_2\to B_1\otimes B_2\) can be regarded as a representation with property (3) and therefore extended to a homomorphism

\[ A_1\overset{\vee}{\otimes}A_2\to B_1\overset{\vee}{\otimes}B_2 . \]

Theorem 2. Suppose that \(u_i\) maps \(A_i\) onto \(B_i\), and denote its kernel by \(I_i\); then \(u\) has kernel \(I_1\overset{\vee}{\otimes}A_2 + A_1\overset{\vee}{\otimes}I_2\). (\(I_1\overset{\vee}{\otimes}A_2\) and \(A_1\overset{\vee}{\otimes}I_2\) denote, respectively, the closures of \(I_1\otimes A_2\) and \(A_1\otimes I_2\) in \(A_1\overset{\vee}{\otimes}A_2\).)

Let \(\omega\) be the natural homomorphism \(A\to A/I\); obviously \(\operatorname{Ker}u\supset I\); it remains to prove that

\[ \|\omega(x)\|\leq \|u(x)\| \quad \text{for } x\in A_1\otimes A_2 . \tag{4} \]

If \(x_i\in A_i\), then \(\omega(x_1\otimes x_2)\) depends only on \(u_1(x_1)\) and \(u_2(x_2)\). Put

\[ \omega(x_1\otimes x_2)=v(u_1(x_1),u_2(x_2)); \]

\(v\) defines a homomorphism \(v':B_1\otimes B_2\to A/I\) with the property

\[ v'(y_1\otimes y_2)=v(y_1,y_2); \]

we shall show that

\[ \|v'(y_1\otimes y_2)\|\leq \|y_1\|\cdot \|y_2\| \quad \text{for } y_i\in B_i, \]

in other words, that

\[ \|\omega(x_1\otimes x_2)\|\leq \|u_1(x_1)\|\cdot \|u_2(x_2)\| \quad \text{for } x_i\in A_i . \]

It is enough to prove that

\[ \|v'(y_1\otimes y_2)\|\geq \|y_1\|\cdot \|y_2\| \quad \text{for } y_i\in B_i, \tag{5} \]

Since \(x_1\otimes x_2-(x_1+z_1)\otimes(x_2+z_2)\in I\), it follows that

\[ \|\omega(x_1\otimes x_2)\|\leq \|(x_1+z_1)\otimes(x_2+z_2)\| =\|x_1+z_1\|\cdot \|x_2+z_2\|. \]

From (5) it follows that

\[ \|v'(y)\|\leq \|y\|_{\nu}\quad \text{for } y\in B_1\otimes B_2, \]

and hence (4) is readily obtained.

Remark 2. Up to now the analogous result has not been proved for \(A_1\overset{*}{\otimes}A_2\); on the other hand, it is known for \(A_1\overset{*}{\otimes}A_2\), but not for \(A_1\overset{\vee}{\otimes}A_2\), that the tensor product of two exact homomorphisms is also exact. From all this one may conclude that, if \(A_1\) or \(A_2\) is of type I, then \(\operatorname{Ker} u=I\), even when \(u_1\) and \(u_2\) are not mappings “onto.”

Remark 3. It is known \((^2)\) that if \(A_1\) and \(A_2\) are simple, then \(A_1\overset{*}{\otimes}A_2\) is also simple; this is unknown for \(A_1\overset{\vee}{\otimes}A_2\).

4. Representations of \(A_1\overset{\vee}{\otimes}A_2\). Let \(\pi_i\) be a representation of \(A_i\) in the Hilbert space \(H_i\). The representation \(\pi_1\otimes\pi_2\) of the algebra \(A_1\overset{*}{\otimes}A_2\) in the space \(H_1\otimes H_2\) was studied in \((^{1,4})\); if it is multiplied on the left by the canonical homomorphism \(A_1\overset{\vee}{\otimes}A_2\to A_1\overset{*}{\otimes}A_2\), one obtains a representation \(\pi_1\overset{\vee}{\otimes}\pi_2\) of the algebra \(A_1\overset{\vee}{\otimes}A_2\), which has some of the properties of \(\pi_1\overset{*}{\otimes}\pi_2\) (the same is true for tensor products of traces \((^1)\)); for example, \(\pi_1\overset{\vee}{\otimes}\pi_2\) is irreducible and completely continuous if and only if \(\pi_1\) and \(\pi_2\) are irreducible and completely continuous \((^1)\); if \(A_1\) and \(A_2\) are assumed separable, then the analogous assertion is true with respect to irreducibility and normality \((^1)\).

It is proved, as in \((^1)\), that every factor representation of finite type (respectively, every irreducible completely continuous representation) of \(A_1\overset{\vee}{\otimes}A_2\) is the tensor product of two representations with the same properties.

It follows from \((^{1,4})\) that \(A_1\overset{\vee}{\otimes}A_2\) is CCR if and only if \(A_1\) and \(A_2\) are CCR; that if \(A_1\) and \(A_2\) are assumed separable, then the same is true with respect to the property GCR; finally, it is proved, as in \((^1)\), that if \(A_1\) and \(A_2\) are separable and \(A_1\overset{\vee}{\otimes}A_2\) is NGCR, then \(A_1\) or \(A_2\) is NGCR; it is unknown whether the converse is true.

Remark 4 (examples of irreducible representations of \(A_1\overset{\vee}{\otimes}A_2\) that are not tensor products of representations). Let \(A_1\) be a \(C^*\)-algebra with a factor representation \(\rho_1\) of type II; \(\rho_1\) is quasi-equivalent to a representation \(\pi_1\) such that the corresponding factor \(B\) is anti-isomorphic to its commutant \(B'\); let \(\Phi\) be an anti-isomorphism of \(B\) onto \(B'\); \(\pi_2(x)=\Phi(\pi_1(x))\) is a representation of the algebra \(A_2\), opposite to \(A_1\); the representation \(\pi\) of the algebra \(A_1\otimes A_2\), defined by equality (2), is irreducible, but is not quasi-equivalent to a tensor product of representations, since its restrictions to \(A_1\) and \(A_2\) are not of type I.

5. The \(C^*\)-algebra of the direct product of groups. Let \(G_1\) and \(G_2\) be locally compact groups, \(A_1\) and \(A_2\) their \(C^*\)-algebras, and \(A\) the \(C^*\)-algebra of the group \(G=G_1\times G_2\).

Theorem 3. There exists an isomorphism of \(A_1\overset{\vee}{\otimes}A_2\) onto \(A\) which sends \(x_1\otimes x_2\) to the function on \(G\): \((g_1,g_2)\mapsto x_1(g_1)\cdot x_2(g_2)\) (\(x_i\) is a continuous finite function on \(G_i\)).

Denote by \(K_i\) the collection of continuous finite functions on \(G_i\); assigning to each element \(x_1\otimes x_2\), \(x_i\in K_i\), the function on \(G\):
\[ (g_1,g_2)\mapsto x_1(g_1)\cdot x_2(g_2), \]
we obtain an isomorphism \(\Phi\) of the algebra \(K_1\otimes K_2\) onto an everywhere dense subalgebra of the algebra \(A\); we must prove that
\[ \|\Phi(x)\|_A=\|x\|_{\vee} \]
for every
\[ x=\sum_{i=1}^{n} x_{1,i}\otimes x_{2,i}\in K_1\otimes K_2. \]
Consider all

pairs \((\pi_1,\pi_2)\) of unitary representations of \(G_1\) and \(G_2\) satisfying the condition

\[ \pi_1(g_1)\cdot \pi_2(g_2)=\pi_2(g_2)\cdot \pi_1(g_1). \]

For such a pair let \(\pi_1 \times \pi_2\) denote the following representation

\[ (\pi_1 \times \pi_2)(g_1,g_2)=\pi_1(g_1)\cdot \pi_2(g_2); \]

then

\[ \|x\|_\nu=\sup\left\|\sum_{i=1}^{n}\pi_1(x_{1,i})\cdot \pi_2(x_{2,i})\right\| =\sup\|(\pi_1\times \pi_2)(\Phi(x))\|=\|\Phi(x)\|_A. \]

Corollary. If the function \(1\) on \(G_1\) (respectively on \(G_2\)) is the limit (in the topology of uniform convergence on every compact subset) of continuous finite positive definite functions, then the norms \(\|\ \|_*\) and \(\|\ \|_\nu\) on \(A_1\otimes A_2\) coincide.

Indeed, let \(l_1,l_2,l\) be the regular representations of \(G_1,G_2,G\); then

\[ \|x\|_*=\|(l_1\otimes l_2)(x)\|=\|l(\Phi(x))\|=\|\Phi(x)\|_A. \]

Received
10 IX 1964

REFERENCES

\(^1\) A. Guichardet, à paraître dans Ann. sci. École norm. supér., \(^2\) M. Takesaki, On the Cross-Norm of the Direct Product of \(C^*\)-Algebras (in press). \(^3\) T. Turumaru, Tohoku Math. J., 4, 242 (1952). \(^4\) A. Wulfsohn, Bull. Sci. Math., 87, No. 1–2, 13 (1963).