Direct sum of C*-representations

Let $A$ be a $C^*$ -algebra and let $\pi_i:A\to B(H_i)$ , $i\in I$ , be $C^*$ -representations. Their direct sum is the representation

\bigoplus_{i\in I}\pi_i:A\longrightarrow B\left(\bigoplus_{i\in I}H_i\right)

defined by

\left(\bigoplus_i\pi_i\right)(a)(\xi_i)_i =(\pi_i(a)\xi_i)_i.

It is well defined because $\|\pi_i(a)\|\leq\|a\|$ uniformly in $i$ . The representation space is the Hilbert direct sum, so its vectors satisfy $\sum_i\|\xi_i\|^2<\infty$ , even when $I$ is uncountable. Products, adjoints, and linear combinations are preserved coordinatewise, so this bounded operator-valued map is again a $C^*$ -representation.

Kernels and faithfulness

The kernel satisfies

\ker\left(\bigoplus_i\pi_i\right)=\bigcap_i\ker(\pi_i).

Consequently, the direct sum is faithful exactly when the family $(\pi_i)$ separates points of $A$ . This observation constructs faithful representations from sufficiently large families and underlies the universal representation.

Nondegeneracy and reducing summands

Each $H_i$ is a reducing subspace for the direct-sum representation. If every $\pi_i$ is nondegenerate, then $\bigoplus_i\pi_i$ is nondegenerate; conversely, a zero or degenerate summand remains visible as a degenerate reducing part. Unitary intertwiners on the summands assemble into a unitary intertwiner of their direct sums.

Examples and distinction

For characters $\chi_1,\ldots,\chi_n$ of a commutative $C^*$ -algebra, $\bigoplus_j\chi_j$ is the corresponding diagonal matrix representation. The construction is not a direct integral: a direct sum uses counting measure and square-summable coordinate families, whereas a direct integral of representations requires measurable fields and ignores changes on null sets.

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §3.4 on direct sums, cyclic representations, and the universal representation.