LETTER TO THE EDITOR

MATHEMATICS

In my article (A. Guichardet, “On tensor products of \(C^*\)-algebras”), published in DAN, vol. 160, no. 5, 1965, the following corrections must be made.

P. 986, lines 6, 7, and 8 should read

\[ \|xy\|_* \leq \|x\|_* \|y\|_*, \qquad \|xx^*\|_* = \|x\|_*^2, \qquad \|x_1 \otimes x_2\|_* = \|x_1\| \cdot \|x_2\|. \tag{1} \]

P. 986, line 11: printed \(A_1 \otimes A_2\), should read \(A_1 \overset{*}{\otimes} A_2\).

Remark 1. It is now known that these norms do not always coincide (see \((^2)\)).

P. 987, line 20: printed \(A_1 \otimes A_2\), should read \(A_1 \overset{\vee}{\otimes} A_2\).

P. 987, line 10 from the bottom should read

\[ \|\nu'(y_1 \otimes y_2)\| \leq \|y_1\| \cdot \|y_2\| \quad \text{for} \quad y_i \in B_i. \tag{5} \]

P. 987, line 6 from the bottom should read

\[ \|\omega(x_1 \otimes x_2)\| \leq \|x_1 + z_1\| \cdot \|x_2 + z_2\| \quad \text{for} \quad z_i \in I_i. \]

P. 987, line 22 should read: …we have the kernel \(I = I_1 \overset{\vee}{\otimes} A_2 + A_1 \overset{\vee}{\otimes} I_2 \ldots\)

Remark 2. It is now known that if \(A_1\) and \(A_2\) are simple (i.e., without closed two-sided ideals), \(A_1 \overset{\vee}{\otimes} A_2\) is not necessarily simple. In fact, in \((^2)\), p. 119, \(C^*\)-algebras \(A_1\) and \(A_2\) are constructed in a Hilbert space \(H\) with the following properties: 1) \(A_1\) and \(A_2\) commute; 2) \(A_i\) generates a factor \(\mathfrak{A}_i\) of type \(\mathrm{II}_1\); 3) the representation \(\sum x_{1,i} \otimes x_{2,i} \to \sum x_{1,i}x_{2,i}\) of the algebra \(A_1 \otimes A_2\) in \(H\) does not have norm \(\leq \|\ \|_*\). Then the analogous representation of the algebra \(\mathfrak{A}_1 \otimes \mathfrak{A}_2\) in \(H\) does not have norm \(\leq \|\ \|_*\), hence the canonical homomorphism \(\mathfrak{A}_1 \overset{\vee}{\otimes} \mathfrak{A}_2 \to \mathfrak{A}_1 \otimes \mathfrak{A}_2\) is not exact, \(\mathfrak{A}_1 \overset{\vee}{\otimes} \mathfrak{A}_2\) is not simple; on the other hand, \(\mathfrak{A}_1\) and \(\mathfrak{A}_2\) are simple.

P. 988, line 8 from the bottom: printed \(A_1 \overset{\vee}{\otimes} A\), should read \(A_1 \overset{\vee}{\otimes} A_2\).

A. Guichardet

CITED LITERATURE

\(^1\) A. Guichardet, Ann. Sci. École Norm. Supér., 81, 189 (1964). \(^2\) M. Takesaki, Tôhoku Math. J., 16, 111 (1964).