TY - JOUR
AU - Christensen, Erik
PY - 2020/09/03
Y2 - 2021/05/13
TI - The block Schur product is a Hadamard product
JF - MATHEMATICA SCANDINAVICA
JA - Math. Scand.
VL - 126
IS - 3
SE - Articles
DO - 10.7146/math.scand.a-121069
UR - https://www.mscand.dk/article/view/121069
SP - 603-616
AB - <p>Given two $n \times n $ matrices $A = (a_{ij})$ and $B=(b_{ij}) $ with entries in $B(H)$ for some Hilbert space $H$, their <em>block Schur product</em> is the $n \times n$ matrix $ A\square B := (a_{ij}b_{ij})$. Given two continuous functions $f$ and $g$ on the torus with Fourier coefficients $(f_n)$ and $(g_n)$ their convolution product $f \star g$ has Fourier coefficients $(f_n g_n)$. Based on this, the <em>Schur product</em> on scalar matrices is also known as the <em>Hadamard product</em>.</p><p>We show that for a C*-algebra $\mathcal{A} $, and a discrete group $G$ with an action $\alpha _g$ of $G$ on $\mathcal{A} $ by *-automorphisms, the reduced crossed product C*-algebra $\mathrm {C}^*_r(\mathcal{A} , \alpha , G)$ possesses a natural generalization of the convolution product, which we suggest should be named <em>the Hadamard product</em>.</p><p>We show that this product has a natural Stinespring representation and we lift some known results on block Schur products to this setting, but we also show that the block Schur product is a special case of the Hadamard product in a crossed product algebra.</p>
ER -